| L(s) = 1 | − 2-s + 3-s + 4-s − 4.33·5-s − 6-s + 2.21·7-s − 8-s + 9-s + 4.33·10-s + 12-s + 13-s − 2.21·14-s − 4.33·15-s + 16-s − 7.21·17-s − 18-s + 0.399·19-s − 4.33·20-s + 2.21·21-s − 2.48·23-s − 24-s + 13.7·25-s − 26-s + 27-s + 2.21·28-s + 4.07·29-s + 4.33·30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s − 1.93·5-s − 0.408·6-s + 0.838·7-s − 0.353·8-s + 0.333·9-s + 1.36·10-s + 0.288·12-s + 0.277·13-s − 0.593·14-s − 1.11·15-s + 0.250·16-s − 1.74·17-s − 0.235·18-s + 0.0917·19-s − 0.968·20-s + 0.484·21-s − 0.518·23-s − 0.204·24-s + 2.75·25-s − 0.196·26-s + 0.192·27-s + 0.419·28-s + 0.757·29-s + 0.790·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9438 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9438 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 - T \) |
| 11 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 5 | \( 1 + 4.33T + 5T^{2} \) |
| 7 | \( 1 - 2.21T + 7T^{2} \) |
| 17 | \( 1 + 7.21T + 17T^{2} \) |
| 19 | \( 1 - 0.399T + 19T^{2} \) |
| 23 | \( 1 + 2.48T + 23T^{2} \) |
| 29 | \( 1 - 4.07T + 29T^{2} \) |
| 31 | \( 1 - 6.26T + 31T^{2} \) |
| 37 | \( 1 + 9.45T + 37T^{2} \) |
| 41 | \( 1 - 4.44T + 41T^{2} \) |
| 43 | \( 1 + 6.80T + 43T^{2} \) |
| 47 | \( 1 - 10.0T + 47T^{2} \) |
| 53 | \( 1 + 3.84T + 53T^{2} \) |
| 59 | \( 1 - 12.1T + 59T^{2} \) |
| 61 | \( 1 - 10.0T + 61T^{2} \) |
| 67 | \( 1 + 3.01T + 67T^{2} \) |
| 71 | \( 1 + 13.3T + 71T^{2} \) |
| 73 | \( 1 + 3.19T + 73T^{2} \) |
| 79 | \( 1 + 3.46T + 79T^{2} \) |
| 83 | \( 1 - 7.35T + 83T^{2} \) |
| 89 | \( 1 + 16.4T + 89T^{2} \) |
| 97 | \( 1 - 12.1T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.47112782304731892118579451609, −7.03615553939767508667192315254, −6.34597252120012735324835961216, −5.03025113652430699301090466834, −4.39641033243929152298489807261, −3.86825555150325702113906198783, −2.99713577408068970044043319635, −2.16051376332897941295336426911, −1.06181684062680195216136785254, 0,
1.06181684062680195216136785254, 2.16051376332897941295336426911, 2.99713577408068970044043319635, 3.86825555150325702113906198783, 4.39641033243929152298489807261, 5.03025113652430699301090466834, 6.34597252120012735324835961216, 7.03615553939767508667192315254, 7.47112782304731892118579451609
